Verify the identity [tex]{a}^{3} + {b}^{3} = ( {a}^{2} – ab \: + {b}^{2} )[/tex]Please answer in steps please About the author Claire
Answer: Algebraic Identity (a3 – b3) = (a – b) (a2 + ab + b2) Objective To verify the identity a3 – b3 = (a – b)(a2 + ab + b2) geometrically by using sets of unit cubes. Prerequisite Knowledge Volume of a cube = (Edge)3 Volume of a cuboid = l x b x h a3 – b3 = (a – b)(a2 + ab + b2) Materials Required A set of 53 plastic or wooden cubes each of dimensions (1 x 1 x 1 unit) Procedure To verify a3 – b3 = (a – b)(a2 + ab + b2). Let a = 3 and b =1. Take 27 cubes and place them to form a stack consisting of a 9 columns, each column consisting 3 cubes [fig. (i)]. Remove one cube from this stack get a stack of 26 cubes (Arrangement I) CBSE Class 9 Maths Lab Manual – Algebraic Identity (a3 – b3) = (a – b) (a2 + ab + b2) 1 Make arrangement II of 26 cubes. This arrangement consists of three stacks. The first stack consists of 18 cubes such as 9 columns of two cubes each. . The second stack consists of 6 cubes such as two rows of three cubes each. Third stack consist of 1 row of 2 cubes. CBSE Class 9 Maths Lab Manual – Algebraic Identity (a3 – b3) = (a – b) (a2 + ab + b2) 2 Observation Since the two arrangements have equal number of cubes (each arrangement has 26 cubes), the total volume in both the arrangements must be equal. Volume of arrangement I Volume of stack in fig. 1(i) = a3 Volume of stack in fig. 1(ii) = b3 ∴Volume of arrangement I = Volume of stack in fig. 1(i) – Volume of stack in fig. 1(ii) = a3 – b3 Volume of arrangement II Volume of the stack in fig. 2 (i) = (a – b) a2 Volume of the stack in fig. 2(ii) = (a – b)ab Volume of the stack in fig. 2 (iii) = (a – b)b2 Total volume of arrangement II = (a – b)a2 + (a – b)ab + (a – b)b2 = (a – b)(a2 + ab + b2). Since number of cubes in arrangement I and II are equal. ∴a3 – b3 = (a – b)(a2 + ab + b2). Reply
Answer:
Algebraic Identity (a3 – b3) = (a – b) (a2 + ab + b2)
Objective
To verify the identity a3 – b3 = (a – b)(a2 + ab + b2) geometrically by using sets of unit cubes.
Prerequisite Knowledge
Volume of a cube = (Edge)3
Volume of a cuboid = l x b x h
a3 – b3 = (a – b)(a2 + ab + b2)
Materials Required
A set of 53 plastic or wooden cubes each of dimensions (1 x 1 x 1 unit)
Procedure
To verify a3 – b3 = (a – b)(a2 + ab + b2). Let a = 3 and b =1.
Take 27 cubes and place them to form a stack consisting of a 9 columns, each column consisting 3 cubes [fig. (i)].
Remove one cube from this stack get a stack of 26 cubes (Arrangement I)
CBSE Class 9 Maths Lab Manual – Algebraic Identity (a3 – b3) = (a – b) (a2 + ab + b2) 1
Make arrangement II of 26 cubes. This arrangement consists of three stacks.
The first stack consists of 18 cubes such as 9 columns of two cubes each. .
The second stack consists of 6 cubes such as two rows of three cubes each.
Third stack consist of 1 row of 2 cubes.
CBSE Class 9 Maths Lab Manual – Algebraic Identity (a3 – b3) = (a – b) (a2 + ab + b2) 2
Observation
Since the two arrangements have equal number of cubes (each arrangement has 26 cubes), the total volume in both the arrangements must be equal.
Volume of arrangement I
Volume of stack in fig. 1(i) = a3
Volume of stack in fig. 1(ii) = b3
∴Volume of arrangement I = Volume of stack in fig. 1(i) – Volume of stack in fig. 1(ii) = a3 – b3
Volume of arrangement II
Volume of the stack in fig. 2 (i) = (a – b) a2
Volume of the stack in fig. 2(ii) = (a – b)ab
Volume of the stack in fig. 2 (iii) = (a – b)b2
Total volume of arrangement II = (a – b)a2 + (a – b)ab + (a – b)b2 = (a – b)(a2 + ab + b2).
Since number of cubes in arrangement I and II are equal.
∴a3 – b3 = (a – b)(a2 + ab + b2).